Stability of smooth solitary waves under intensity--dependent dispersion

The cubic nonlinear Schrodinger equation (NLS) in one dimension is considered in the presence of an intensity-dependent dispersion term. We study bright solitary waves with smooth profiles which extend from the limit where the dependence of the dispersion coefficient on the wave intensity is negligible to the limit where the solitary wave becomes singular due to vanishing dispersion coefficient. We analyze and numerically explore the stability for such smooth solitary waves, showing with the help of numerical approximations that the family of solitary waves becomes unstable in the intermediate region between the two limits, while being stable in both limits. This bistability, that has also been observed in other NLS equations with the generalized nonlinearity, brings about interesting dynamical transitions from one stable branch to another stable branch, that are explored in direct numerical simulations of the NLS equation with the intensity-dependent dispersion term.